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Chicago Public Schools, Advanced Algebra Lesson plans, Days 37-53

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• 134

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG,CAS,CFS): 6A1

TAP:Analyze and interpret data presented in charts,

graphs, tables and other displays

ISAT:Identify, analyze, and solve problems using

equations, inequalities, functions, and theirgraphs

Unit Focus/Foci

Matrices and Systems of Equations

Instructional Focus/Foci

Using Matrices to Store and Represent Data and Adding or Subtracting Matrices

Materials

Educational Strategies/Instructional Procedures

Give the following definitions:

Matrix: A rectangular arrangement of elements in rows and columns enclosed by brackets.

Example: M = 1 4 3 2

3 2 0 5

LNM

OQP

Matrix notation: An entry is named as mab, where a = row number and b = column number

In the example above, m13= 3; m24=5Dimension: the number of rows x the number of columns gives the dimension of amatrix. The dimension of the matrix M above is 2 x 4.

Have students place elements in their proper position, then have them enter the followingmatrices into their calculators:

• 135

To enter a matrix into the graphing calculator:1. press MATRX2. use the right arrow to EDIT, then ENTER3. enter number of rows, ENTER4. enter number of columns, ENTER5. enter each element, going across. After each element, press ENTER. If you

make a mistake, scroll to the location you want to change.

To perform operations on matrices, press MATRX, leave NAMES highlighted and scroll downto the matrix you want to use; press ENTER.

Students will find: A + B; B A; A A (call this C); C + B; 1 x B

Students will report findings to the whole class.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

Connection(s)

Enrichment: In a recent survey, 90% of high school students were opposed to a systemwideSlice Girls concert suggested by elementary school students.1. Work with a partner to conduct a survey in your school regarding the questions below.

A. Should the Slice Girls concert occur?B. Should high school students be mandated to attend?C. Should elementary school students be mandated to attend?D. What percent have no opinion?

Use the following matrix to report your data in percentage form.

Question Yes No Indifferent

A B=

L

NMMM

O

QPPP

=

L

NMMM

O

QPPP

3 2 1 0

6 2 4 1

0 3 2 1

7 3 2 1

4 0 3 2

1 7 3 0

ABCD

• 136

Fine Arts:

Home: Parents will sign homework record weekly.

Remediation:

Technology:

Assessment

Have students answer the following questions:

a. What must be true about the dimensions of two matrices that are equal? b. Given matrices A, B, and A + B, why must they have the same dimension? c. What is the relationship between matrices A and 1A? d. What is a good name for matrix C?

Homework

Assign appropriate problems from your text.

Teacher Notes

Solutions to Assessment:

a. They must have the same number of rows and columns.b. The entries from the first matrix should correspond to the entries from the second matrix.c. Matrix -1 x A is the opposite of matrix A.d. Zero matrix

Solutions to in-class assignment:

A + B =

11101

17210

11110

B A =

1541

3122

1354

A A =

0000

0000

0000

C + B =

0371

2304

1237

-1 x B

0371

2304

1237

• 137

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG,CAS,CFS): 6A1

TAP:Perform arithmetic operations involving

integers, fractions, decimals and percentsexplicitly stated or within context

Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations

Understand number systemsUse variables, number sentences, and equations

to represent solutions and solve problems

ISAT:Solve problems requiring computations with

whole numbers, fractions, decimals, ratios,percents, and proportions

Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs

Unit Focus/Foci

Exploring Matrices and Systems of Equations

Instructional Focus/Foci

Performing Matrix Multiplication

Materials

Classroom set of graphing calculatorsOverhead graphing calculatorsClassroom set of graphing calculatorsOverhead projector

Educational Strategies/Instructional Procedures

Explain to students: To multiply two matrices Amxn and Bpxq, n must equal p (the inner

dimensions). The resulting matrix AB will have dimension m x q (the outer dimensions). If np, the two matrices cannot be multiplied.

Example 1: A B=

LNM

OQP =

L

N

MMMM

O

Q

PPPP3 2 0 1

4 3 3 2

0

3

2

1

• 138

dim[A] = 2x4, dim[B] = 4x1, so these two matrices can be multiplied (since 4=4), and theresulting matrix will have dimension 2x1.

Take row 1 of A and column 1 of B, multiply successive elements, add these products, and theresult is the corresponding element in the product matrix. Using the example above:

30 + 23 + 02 + 11 = 5; 40 +03 + 32 + -21 = 4. So the product matrix is 5

4

LNMOQP

Give students several matrices of different dimensions and ask if they can be multiplied.

Example 2:

a.

40

36

25

68b.

82

43

96

41

72c.

036

894

282

413

Solutions: a. yes b. yes c. no

Have students form small groups; use the graphing calculator to determine whether theassociative, commutative, or distributive properties hold for matrix multiplication.

Report findings to class. Use the matrices: A =

06

14 B =

63

12 C =

20

31

a. A*B = B * Ab . A(B*C) = (A * B ) * Cc . A(B+C) = AB + AC

Example 3:

Jane and Andrew enjoy shopping for music on a monthly basis. They each purchase thefollowing items.

CDs Tapes LPsJane 10 6 1

Andrew 8 2 3

If CDs cost \$15, tapes cost \$8, and LPs cost \$10, use matrices to find the total cost that Jane andAndrew each spent on music.

• 139

Solution:

328

1610

10

8

15

=

166.

208

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

Connection(s)

Enrichment: Use graph paper to draw squares with areas of 2, 5, and 10 units.

Fine Arts:

Home: Parents will sign homework record weekly.

Remediation:

Technology: Have students review how their calculator can perform various operations withmatrices.

Assessment

Teacher observation

Homework

Assign appropriate problems from your text.

Teacher Notes

This is a two-day lesson. The second day (Day 39) should be used to review homework, forremediation exercises, or for extra practice, as necessary.

Solutions to in-class assignment:

a. A*B = B *A (not true) b. A(B*C) = (A *B) * C (true)

612

25

=

348

214

2412

195=

2412

195

c. A(B+C) = AB + AC (not true)

invalid dimensions =

246

121

• 140

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG,CAS,CFS): 8D2

TAP:Perform arithmetic operations involving

integers, fractions, decimals and percentsexplicitly stated or within context

Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations

Understand number systemsUse variables, number sentences, and equations

to represent solutions and solve problems

ISAT:Solve problems requiring computations with

whole numbers, fractions, decimals, ratios,percents, and proportions

Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs

Unit Focus/Foci

Exploring Matrices and Systems of Equations

Instructional Focus/Foci

Identifying Systems of Two Linear Equations

Materials

Educational Strategies/Instructional Procedures

Students will: Form groups of 2. Complete Handout 4.3 (exploration on intersections of lines). Report findings to the class.

Present the following definitions:

Consistent system: lines intersect in one or more pointsIndependent system: lines intersect in one unique point (one solution)Dependent system: lines intersect in an infinite number of points (the lines coincide; infinitelymany solutions)Inconsistent system: lines do not intersect (parallel lines; no solutions)

• 141

Give examples of systems of equations and graphs of solutions of equations.

Have students match each system with its graph.

a. y = x + 1 b. y = -2x + 3 c. y = .5x - 2 y = x y = 3x + 1 y = -x + 4

1. 2.

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